Stability of Vector Functional Differential Equations: A Survey

Abstract

This paper is a survey of the recent results of the author on the stability of linear and nonlinear vector differential equations with delay. Explicit conditions for the exponential and absolute stabilities are derived. Moreover, solution estimates for the considered equations are established. They provide the bounds for the regions of attraction of steady states. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the following methods and results: a) the generalized Bohl-Perron principle and integral version of the generalized Bohl-Perron principle; b) the freezing method; c) the positivity of the fundamental solutions. A part of the paper is devoted to the Aizerman-Myshkis problem in the the absolute stability theory and to integrally small perturbations of stable equations.

Quaestiones Mathematicae 35(2012), 83–131.